Object selection box with eyedropper
I spoke to Campbell Barton (the developer who got eyedropper functionality implemented) about this, he said that this kind of functionality is not available in Python API at the moment.

UPDATE:

Blender 2.79 now allows to create add-on properties which reference datablocks.  

custom_property = PointerProperty(type=bpy.types.Object)


Using this property with a prop_search() we'll get search box with an eyedropper.

UPDATE 2:

Usage of prop_search() is not necessary, in this case prop() will do just fine.
Noted by J. Bakker.